A \emph{shell} is a structure $\mathbf{S}=\langle S,+,0,\cdot,1 \rangle$ of type $\langle 2,0,2,0\rangle$ such that

$0$ is an identity for $+$: $0+x=x$, $x+0=x$

$1$ is an identity for $\cdot$: $1\cdot x=x$, $x\cdot 1=x$

$0$ is a zero for $\cdot$: $0\cdot x=0$, $x\cdot 0=0$

Let $\mathbf{S}$ and $\mathbf{T}$ be shells. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$, $h(1)=1$

Example 1:

$\begin{array}{lr} f(1)= &1 f(2)= & f(3)= & f(4)= & f(5)= & f(6)= & \end{array}$

Semirings with identity and zero